Evolution of Controllability in Interbank Networks Danilo Delpini1,2 , Stefano Battiston3 , Massimo Riccaboni2,4 , Giampaolo Gabbi5,6 , Fabio Pammolli2,7 , and Guido Caldarelli∗2,8,9 1
Department of Economics and Business - DiSEA, University of Sassari, via Muroni 35, 07100
Sassari, Italy 2
IMT Institute for Advanced Studies, Piazza San Ponziano 6, 55100 Lucca, Italy
3
Chair of Systems Design - MTEC, ETH Zurich, Weinbergstrasse 56-58, 8092 Zurich, Switzerland
4
Department of Managerial Economics, Strategy and Innovation, Katholieke Universiteit Leuven,
3000 Leuven, Belgium 5
University of Siena, Department of Management and Law, Piazza S. Francesco, 53100 Siena,
Italy 6
SDA Bocconi School of Management, via Bocconi 8, 20136 Milan, Italy
7
Center for Polymer Studies and Department of Physics, Boston University Boston, MA 02115
8
Istituto dei Sistemi Complessi (ISC), Department of Physics, Universit`a ”Sapienza”, Pl. Aldo
Moro 2, 00185 Roma, Italy 9
London Institute for Mathematical Sciences, 35a South St. Mayfair W1K 2XF London UK
*
Correspondence and requests for materials should be addressed to G.C. (
[email protected])
Supplementary information for the article “Evolution of Controllability in Interbank Networks”
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I.
The network of interbank money market
I. A
Data set
Our data set is composed of 2750 daily snapshots of the Italian interbank money market, from January 4th, 1999 to September 30th, 2009. This data has been collected by “e-MID”, the only electronic market for Interbank Deposits in the Euro Area and US 1 . For the most part, the transactions correspond to overnight exchanges of deposits among banks. The amounts of transactions between parties, as well as their directions are recorded individually on a daily basis. Generally, any two banks i and j exchange liquidity in multiple tranches in both directions.
All these daily transactions form an oriented graph G(1) (Vt , Et ), with t = 1, 2, . . . 2750, whose vertices are the banks taking part in the deals and the links have a weight given by the corresponding cash flow. We name Vt and Et the sets of nodes and edges respectively, their cardinalities |Vt | , |Et | giving the number of nodes and edges of the graph. All these quantities depend on time because the number of banks entering and exiting the market as well as the amounts exchanged change considerably. In particular, we denote wt (i, j) the aggregate lending of bank i to j on day t. It is worth mentioning that we may well have also j lending to i, corresponding to wt (j, i) 6= 0, that is flows are generally bidirectional. Fig. S1 illustrates the evolution of G(1) in terms of number of nodes, number of edges and total lending volumes (the sum of all link weights, indicated as vt ). 1
e-MID Sim S.p.A, Milan, Italy.
2
2.0
|V | / |V0 | |E| / |E0 | v/v0
1.5 1.0 0.5 0.0
1999
2001
2003
2005
2007
2009
Figure S1: Evolution of the daily interbank lending network G(1) (V, E). The system experimented a sensible shrinking in both the number of agents (|V |) and the number of links (|E|); a credit crunch process is also clear starting in 2007, with a volume (v) drop of roughly 80% with respect to 1999. Curves are rescaled by the values of the first point of the corresponding quantity |V0 | , |E0 | , v0 respectively.
I. B
Network aggregation
As we can see from Fig. S1, the size of the network shrinks steadily over time, losing about half of the nodes and links. In contrast, the lending volume increases by 50% from 1999 to 2007. Afterwards, it experiences a fast drop between 2007 and 2009, decreasing by about 80%. So, we see that the network under examination undergoes important changes in terms of size and volumes flowing through the agents. However, some topological properties related to the ”fitness” of vertices seem to be preserved despite the shrinking process. For instance, the network remains scale free throughout the whole period, as shown in1 . Remarkably, the density of links remains also around similar values (2.8% in 1999, 2.3 % in 2005, again 2.8% in 2009).
On a daily basis, the set of transactions Et is highly volatile. However we can aggregate
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the daily loans over a number of trading days ∆, and define links and weights at this time scale as wu,∆ (i, j) =
P∆−1 s=0
wu+s (i, j), where u = 1, 1+∆, 1+2∆, . . . . In this way, we obtain the network
G(∆) (Vu,∆ , Eu,∆ ), which is representative of the banking system during the time span [u, u + ∆). In our empirical study we compare results for five different time scales ∆ = 1, 5, 21, 63, 126 days, roughly corresponding to daily, weekly, monthly, three and six months time scales. It is worth noting that the recent Basel Committee proposals (Basel III) includes a 30-day liquidity coverage ratio requirement whose purpose is to ensure banks maintain adequate levels of unencumbered high quality assets against net cash outflows under stress.
I. C
Centrality measures
The systemic relevance of banking institutions is key when designing regulation and monetary policies2, 3 . A reasonable assumption is to consider the case of the “too connected to fail”, even though the precise notion of systemic relevance has not been defined yet and different measures have been proposed4–6 . The degree k(i) of a vertex provides information about the centrality in terms of number of connections to other nodes. This feature determines the properties of resilience of the network against link failure and epidemic spreading7, 8 . Another measure of importance is the closeness centrality. We introduce a notion of distance as follows. First we define the average lending w¯u of the graph as the average weight of its edges. This represents the average amount of money which is exchanged between two banks on that period and provides a convenient measure unit. Given two connected nodes i → j, we define their distance as the inverse of the
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(normalized) edge weight d(i, j) = 1/(w(i, j)/w), ¯ so that the more i is lending to j, the closer is i to j. In this respect, this distance is a proxy of how much j is exposed towards i. For an oriented shortest path (k1 → k2 → · · · → kn−1 → kn ) connecting k1 to kn we define the distance as d(k1 , kn ) =
Pn−1 s=1
d(ks , ks+1 ). The closeness centrality ct (i) of a vertex is defined as the inverse of
the average distance from the other nodes 2 . It measures the strengths of the credit relations among a bank i and its direct and indirect borrowers, also providing an indicator of potential spillover due to inefficient liquidity provision from bank i. In principle (but also in practice) we may have banks with very small degree (few connections) but with very tight links (large loans) or hub nodes which exchange very small amounts with many banks.
II.
Network controllability
II. A
General framework
It is possible to gain control of a directed network by identifying the “driver” nodes through the Minimum Input Theorem9 : the minimum number of nodes to be controlled (drivers) corresponds to the number of uncovered nodes in a maximum matching of the corresponding graph.
For directed graphs a matching M ⊆ E is a subset of directed edges such that any two edges do not share the same start or end vertex (note that for two edges in M , the end vertex of one may well be the start vertex of the other). The maximum matching M max is the one with the largest possible cardinality among all the graph’s matchings. A vertex is uncovered or unmatched when 2
To account for changes in the network shape, we normalize closeness to (|V | − 1)/(|E| − 1).
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no edge in M max reach it.
It has to be noted that, in general, a large multiplicity of maximum matchings exists. In the scope of the analysis presented in the main text, we always select the matching for which the sum of its edges’ weights is maximal. Indeed, a maximum matching provides a maximal set of paths where some nodes are controlled indirectly (i.e. without external inputs) by their predecessors. Through these paths, the effect of external controllers on the drivers is propagated to other nodes. More precisely, if node i controls node j, the time variation of the matched node’s state xj gets a contribution from i that is proportional to the edge weight wij , see Eq. (1). In this sense, the larger the weight, the larger the impact of i upon j and, intuitively, maximising the sum of weights (all weights are positive in our case) would mean maximising the sum of the impacts.
The above theorem builds on a linear dynamics of the form dx(t) = Ix(t) + Cu(t) dt
(1)
where xi=1,...,N is a state variable for the N nodes, I is an N × N “influence” matrix representing the way every node is influenced by its predecessors. The components of the vector u correspond to external functions that are applied to a subset of ND nodes and C is an N × ND “control” matrix of external weights. The structure of I corresponds to that of the transposed weighted adjacency matrix (see9 for more details), so that the time variation of xi is determined, through I, by the state of the nodes that points to i. Generally, the exact values of the entries in I are unobservable and only the matrix structure is required in order to apply the theorem.
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For the interbank lending network, we identify xi with the aggregate lending of bank i, i.e. the sum of its outgoing lending flows, and we assume that the arrows of the mutual influence relationships among banks correspond to the observed flows. Indeed, it is reasonable to assume that variations of the amount of lending from any bank depend on the lending of the others 10–12 . Moreover, the information a given bank is able to access concerns only the level of lending it receives from its neighbors while information about all other deals is usually confidential. In particular, during a credit crunch, a bank gets less credit from its counterparts; as a reaction, that bank is likely to lower its lending to the others generating dangerous cascade effects. Even though this dependence may have a quite complicated formalization, as a first order approximation we will take it linear, so that we can postulate a dynamics of the form (1) for x(t), the functions u(t) representing external actions from regulators.
Now let us focus on the fraction of drivers nD = |D| /N , where D = {ik1 , ik2 , . . . , ik|Dt | } is the set of the drivers given M max . A large value of nD would indicate that the interbank system is problematic from a control perspective, since many banks would be responsible for variations of the state of whole network. Conversely, regulators and policy makers would appreciate small values of nD . Indeed, in this case it is enough to intervene only in a restricted number of banks.
II. B
Effect of network aggregation
For temporal networks, the fraction of drivers increases, in general, in the process of aggregating the edges of ∆ consecutive instances of the graph. However, the exact scaling of nD vs ∆ is 7
difficult to predict. For the interbank lending system, our analysis detects a neat power-law scaling (see Fig. 1 (c) in the main text), and we may wonder if this law is valid any times we add graph instances and independently on the specific dynamics underlying the graph evolution. General arguments, as developed below, show that this is not the case, at least when graph instances are independent.
First, consider a sequence of independent Erd¨os-Renyi graphs with N nodes and linking probability p and the graph G(∆) obtained by aggregating the links of ∆ instances (for the sake of clarity, we limit the discussion to unweighted graphs.). It can ben shown that the average degree of G(∆) would be hki(∆) = 2p(∆)(N −1), with p(∆) = [1−(1−p)∆ ], and the graph rapidly becomes complete with nD approaching its lower limit 1/N . In 9 it has been shown that nD is approximately an exponential function of hki, both for random and scale-free graphs. In our example we would ∆
then expect nD ≈ e−c(1−p) , with c constant, rather than a power-law decay.
In the case of identical, independent scale-free graphs, we haven’t got a closed-form expression for hki(∆) . For large hki, we would expect log nD ≈ −c
Pkmax k=0
∆ k(Fk∆ − Fk−1 ), where Fk is
the cumulative distribution function of the nodes’ degree evaluated in k. To get an insight of what scaling should be expected, we simulate H = 100 instances of a scale-free graph with N = 200 nodes and average degree approximately 6. These numbers correspond (approximately) to the average number of active nodes and their average degree for the interbank lending system. Graphs are generated from the static model
13
with in-degree and out-degree tail exponents equal to 2.7
and 2.15 respectively, as observed empirically in 1 . In Fig. S2 we plot the scaling of nD vs ∆.
8
Average nD
100 10-1 10-2 10-3 0 10
∆
101
Figure S2: Average fraction of drivers for a network obtained aggregating ∆ independent instances of a scale-free graph. Instances are generated from the static model with N = 200 nodes, instance’s average degree hki = 6, in-degree and out-degree tail exponents γin = 2.7 and γout = 2.15 respectively.
Comparing with Fig. 1 (c) in the main text, here we see a rather different scaling; in particular, when adding independent scale-free graphs, the aggregated network saturates at a much higher speed than for the interbank system.
We conclude that the scaling we observe empirically is a genuine feature of the dynamics and topology of the interbank lending network. In particular, the process of aggregation involves instances that can not be considered realizations of the same graph and, furthermore, the underlying financial dynamics is likely to generated correlations among those instances.
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II. C
Driver persistence
For a temporal network, like the interbank lending one, a critical aspect is represented by the stability of the set of drivers with respect to time. In order to investigate this issue, we define the drivers’ resilience as the average fraction of drivers which persists in the network control set after a time lag τ (in days) rD,τ =
Du+τ ∩ Du | Du |
(2)
where the average is taken with respect to the time index u.
References
1. De Masi, G., Iori, G. & Caldarelli, G. Fitness model for the Italian interbank money market. Phys. Rev. E 74, 066112 (2006). 2. Stern, G. H. & Feldman, R. J. Too Big To Fail (Brookings Institution Press, Washington, 2004). 3. Freixas, X. Monetary policy in a systemic crisis. Oxford Review of Economic Policy 25, 630–653 (2009). 4. Battiston, S., Gatti, D. D., Gallegati, M., Greenwald, B. & Stiglitz, J. E. Default cascades: When does risk diversification increase stability? Journal of Financial Stability 8, 138–149 (2012).
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5. Battiston, S., Delli Gatti, D., Gallegati, M., Greenwald, B. & Stiglitz, J. E. Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk. J. Econ. Dyn. Control 36, 1121–1141 (2012). 6. Battiston, S., Puliga, M., Kaushik, R., Tasca, P. & Caldarelli, G. DebtRank: too central to fail? Financial networks, the FED and systemic risk. Sci. Rep. 2, 541 (2012). 7. Pastor-Satorras, R. & Vespignani, A. Epidemic Spreading in Scale-Free Networks. Phys. Rev. Lett. 86, 3200–3203 (2001). 8. Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003). 9. Liu, Y.-Y., Slotine, J.-J. & Barab´asi, A.-L. Controllability of complex networks. Nature 473, 167–73 (2011). 10. Acharya, V. V. & Merrouche, O. Precautionary Hoarding of Liquidity and Inter-Bank Markets: Evidence from the Sub-prime Crisis. NBER Work. Pap. Ser. No. 16395 (2010). 11. Heider, F., Hoerova, M. & Holthausen, C. Liquidity Hoarding and Interbank Market Spreads: The Role of Counterparty Risk (2009). 12. Galbiati, M. & Sor¨amaki, K. Liquidity-saving mechanisms and bank behaviour. Bank of England Working Paper No. 400 (2010). 13. Goh, K.-I., Kahng, B. & Kim, D. Universal Behavior of Load Distribution in Scale-Free Networks. Phys. Rev. Lett. 87, 278701 (2001).
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